Sum ratio

Here is a puzzle magic trick to amaze your friends

Puzzle

The magician asks a volunteer to secretly pick any two 2-digit numbers and add them together. The the magician instructs the volunteer to take the largest two numbers from the three numbers (the original 2 and the sum) and add them together. Again, the volunteer is to take the largest two numbers (of the four now) and add them. The volunteer keeps doing this as often as they want, always adding the largest two numbers. When the volunteer is satisfied, there is one more task. They must take the largest number and divide it by the second largest. The magician, without ever looking at any of the numbers, correctly identifies the ration as approximately 1.61803398...

How?